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The basic hypothesis of a post-Copernican cosmological theory is that all

the points of the universe have to be essentially equivalent. This

hypothesis is required in order to avoid any privileged observer.

This assumption has been implemented by Einstein in the s.c. *cosmological

principle*: all the positions in the universe have to be essentially

equivalent, so that the universe is (at least mathematically) *homogeneous*.

This situation implies the condition of some (spherical) symmetry about

every point, so that the universe is (at least locally) *isotropic*.

But an *hidden* assumption seems to be in the formulation of the

cosmological principle. In fact, the condition that all the points are

(statistically) equivalent (with respect to their environment) corresponds

to the property of a local *isotropy*. And it is generally accepted that the

universe can not be *isotropic* about every point without being also

*homogeneous*.

But local *isotropy* does not necessarily implies *homogeneity*. In fact a

topological theorem states that homogeneity requires (at least local)

isotropy together with the assumption of the *analyticity*. Analyticity was

an usual assumption in any physical problem: before the *fractal* geometry!

Actually a *fractal* structure has some local isotropy but has not

homogeneity. In simple terms one observes the same mix (structures and

vacua) in different directions (statistical isotropy). This means that a

*fractal* structure satisfies the cosmological principle! In the sense that

all the points are essentially equivalent (no center, no special points).

But this does *not* imply that these points are distributed uniformly!

Now astronomy showed some intrinsically *irregular* structures for which the

analyticity assumption might be reconsidered and fractal properties might be

investigated.

The space distribution of galaxies and clusters, the cosmic microwave

background radiation, the linearity of the redshift-distance relation

(Hubble law), the abundance of (light) elements in the universe: each of

these four points provides independent experimental facts. The objective of

a cosmological theory of the universe (fractal or not) should be to provide

a coherent explanation of all these facts together. An important point in

the theoretical investigation concerns the distribution of the gravitational

force inside structures which could be irregular or fractal.

But the recent statistical analysis of the experimental data already shows

also that *the distribution of galaxies is fractal* up to the deepest

observed scales. In the near future one could describe structures in which

intrinsic *self-similar* irregularities develop at *all* scales and

fluctuations cannot be described in terms of *analytical* functions. The

theoretical methods to describe this situation could not be based on

ordinary differential equations because *self-similarity* implies

singularities and the absence of analyticity.

About the fractal universe:

http://pil.phys.uniroma1.it/astro.html

http://pil.phys.uniroma1.it/debate.html

http://pil.phys.uniroma1.it/

http://pil.phys.uniroma1.it/eec1.html

The Nobel laureate (1977) P.W. Anderson is working on this field (now at the

Princeton University and also at the Rome University, La Sapienza).

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